Graded Kronecker Modules of Constant Rank

Let $K(n)$ be the generalized Kronecker quiver, and let $(T(n),\mathrm{\Omega })$ be the universal covering of $K(n)$, where $n\ge 3$ and $\mathrm{\Omega }$ is a bipartite orientation. We introduce three different types of representations for quiver $(T(n),\mathrm{\Omega })$ by $K(n)$: maps are injective in a representation maps are all surjective in a representation, and representations are of constant rank. We use $\text{Inj},\text{Sur}$ and $\text{GR}$ to denote the corresponding subcategories containing these representations, respectively, and we show that \begin{center} $\text{GR}=\text{Inj}\cup \text{Sur}$. \end{center}